6.2 Double Exchange
185
but the matrix element between Φ 1 and Φ 4 is slightly more involved
Φ 1 | ˆ
H|Φ 4 ==a 1 b 1 a 2 | ˆ
H|a 1 b 1 b 2 ==a 2 | ˆ
h|b 2 ++a 1 a 2 |
1 − ˆ
P 12
r 12
|a 1 b 2
++b 1 a 2 |
1 − ˆ
P 12
r 12
|b 1 b 2 ==a 2 | ˆ
h|b 2 ++a 1 a 2 |
1
r 12
|a 1 b 2 ++b 1 a 2 |
1
r 12
|b 1 b 2
(6.18)
where the two-electron integrals cannot be written as Coulomb or exchange integrals.
The sum of the three terms can be considered as the hopping parameter t, similar to
the expressions given in Eqs. 6.1b and 6.1c. The complete interaction matrix is
ˆ
H
|Φ 1 | Φ 2 | Φ 3 | Φ 4 | Φ 5 | Φ 6
Φ 1 |
−K ′
0
−Kt
00
Φ 2 |
0
−K
−K ′
00t
Φ 3 |
−K
−K ′
00t
0
Φ 4 |
t
00−K ′
0
−K
Φ 5 |
00t
0
−K
−K ′
Φ 6 |
0t
0−K
−K ′
0
K = K a 1 a 2 = K b 1 b 2 is the on-site exchange interaction and K ′ = K a 1 b 1 is the
intersite exchange. Two approximations have been made to obtain this matrix. In the
first place, it is assumed that K a i b j with i = j can be neglected. Furthermore, we
assume that the effect of the so-called singlet displacement operator is small enough
to be omitted. The action of this operator is illustrated in Fig. 6.5 and transforms Φ 1
into Φ 5 or Φ 6 , and Φ 4 into Φ 2 or Φ 3 .
6.3 Show that the other zeros in the matrix are real zeros and not due to any
additional approximation. Explain why in the Hamiltonian matrix H 11 = H 44 ,
H 22 = H 55 , H 33 = H 66 , and H 13 = H 46 .
The diagonalization of the matrix yields six eigenstates, two quartets and four
doublets with the following energies after shifting all states with K + K ′ to let the
zero of energy coincide with the average energy of the quartet states
Fig. 6.5 Action of the
singlet displacement
operator
185
but the matrix element between Φ 1 and Φ 4 is slightly more involved
Φ 1 | ˆ
H|Φ 4 ==a 1 b 1 a 2 | ˆ
H|a 1 b 1 b 2 ==a 2 | ˆ
h|b 2 ++a 1 a 2 |
1 − ˆ
P 12
r 12
|a 1 b 2
++b 1 a 2 |
1 − ˆ
P 12
r 12
|b 1 b 2 ==a 2 | ˆ
h|b 2 ++a 1 a 2 |
1
r 12
|a 1 b 2 ++b 1 a 2 |
1
r 12
|b 1 b 2
(6.18)
where the two-electron integrals cannot be written as Coulomb or exchange integrals.
The sum of the three terms can be considered as the hopping parameter t, similar to
the expressions given in Eqs. 6.1b and 6.1c. The complete interaction matrix is
ˆ
H
|Φ 1 | Φ 2 | Φ 3 | Φ 4 | Φ 5 | Φ 6
Φ 1 |
−K ′
0
−Kt
00
Φ 2 |
0
−K
−K ′
00t
Φ 3 |
−K
−K ′
00t
0
Φ 4 |
t
00−K ′
0
−K
Φ 5 |
00t
0
−K
−K ′
Φ 6 |
0t
0−K
−K ′
0
K = K a 1 a 2 = K b 1 b 2 is the on-site exchange interaction and K ′ = K a 1 b 1 is the
intersite exchange. Two approximations have been made to obtain this matrix. In the
first place, it is assumed that K a i b j with i = j can be neglected. Furthermore, we
assume that the effect of the so-called singlet displacement operator is small enough
to be omitted. The action of this operator is illustrated in Fig. 6.5 and transforms Φ 1
into Φ 5 or Φ 6 , and Φ 4 into Φ 2 or Φ 3 .
6.3 Show that the other zeros in the matrix are real zeros and not due to any
additional approximation. Explain why in the Hamiltonian matrix H 11 = H 44 ,
H 22 = H 55 , H 33 = H 66 , and H 13 = H 46 .
The diagonalization of the matrix yields six eigenstates, two quartets and four
doublets with the following energies after shifting all states with K + K ′ to let the
zero of energy coincide with the average energy of the quartet states
Fig. 6.5 Action of the
singlet displacement
operator